Nash Equilibrium in Strategic Bidding: a Binary Expansion Approach

The objective of this work is to present a binary expansion (BE) approach to the calculation of Nash equilibrium in electricity markets. The BE scheme is used to transform the products of variables in the nonlinear bidding problem into a mixed integer linear (MILP) formulation, which can be solved by commercially available computational systems. The BE scheme is flexible, and applicable to applicable to Cournot, Bertrand or joint price/quantity bidding models. It is also possible to represent transmission networks (zonal and DC power flow), uncertainties (scenarios for plant availability and load) and unit commitment. We start from the basic equilibrium formulation: no agent can improve its revenues by changing its bid, given the equilibrium bids of the remaining agents. Next, we show that, for a discrete set of bidding decisions (which is the case of the BE formulation) the no-improvement condition for each agent can be handled by enumeration. In this case, the Nash equilibrium conditions become a set of simultaneous MILP constraints, which can be solved by commercially available software. It is also possible to systematically generate the multiple Nash equilibria. The application of the methodology is illustrated in case studies with configurations derived from the 80 thousand-MW Brazilian system.

Supply Adequacy in the Brazilian Power Market

This paper discusses how the issue of supply adequacy is treated in the Brazilian power market, analyzing lessons learned from past experiences and describing the solutions that are being implemented to reconcile competition in generation with an adequate supply.

We extend a static mixed integer disjunctive (MID) transmission expansion planning model so as to deal with circuit contingency criterion. The model simultaneously represents the network constraints for base case and each selected circuit contingency. The MID approach allows a commercial optimization solver to achieve and prove solution optimality. The proposed approach is applied to a regional Venezuelan network, and results are discussed.

Transmission Expansion Planning In The Western Interconnection – The Planning Process and the Analytical Tools That Will Be Needed to do the Job

Within the Western Interconnection there is a growing realization that the traditional long-term planning processes and analytical methods used to evaluate wires and non-wires alternatives fail to address many of the questions arising from deregulation and the perceived need for new infrastructure to facilitate markets and to assure reliability. In addition, homeland security issues may influence transmission infrastructure expansion. This paper describes the region’s evolving transmission expansion planning process and the role OPF modeling plays in aiding decision-making and consensus building. It identifies some opportunities for improvement in modeling, highlights two state-of-the-art models with capabilities that go beyond those of other models currently used for long-term transmission planning and makes a case for why we need to invest in better modeling and databases.

Transmission Structure in Brazil: Organization, Evaluation and Trends

This paper describes the structure and regulation of transmission-related activities in Brazil, and assesses their effectiveness/limitations when addressing issues such as network expansion, locational pricing, transmission financial rights, and others. The paper also discusses the cross-border energy trading between Brazil and neighbor countries, and the perspectives of an integrated regional electricity market.

The objective of this paper is to present an optimization model for hydro scheduling and risk management in deregulated electricity markets where hydropower companies are likely to face stochastic inflows, spot prices and forward prices. This portfolio management problem, which includes physical and financial assets, is formulated as a stochastic revenue maximization problem for a given risk profile. The model seeks to maximize company’s revenues by simultaneously determining generation, selling and purchasing decisions on both the spot and the forward markets. The company’s risk aversion is apprehended by penalizing release and contracting policies that lead to unacceptable financial performances. A hybrid stochastic dynamic programming (SDP)/stochastic dual dynamic programming (SDDP) formulation is adopted to solve this large-scale optimization problem. The state variables of the SDP/SDDP model are the volume in storage at the beginning of each time period, the previous inflows, the accumulated balance of forward contracts, the accumulated revenues and the accumulated profit for each profit period. Spot price and hydrologic uncertainties are captured through scenarios. The model also used a Markov chain description for forward prices. Market liquidity must be defined for each financial product. A hydropower company in the French market is used to illustrate this integrated physical and financial planning model.

This paper presents an optimization procedure for transmission expansion planning that takes into account multiple dispatch scenarios, contingency constraints and multiple stages. The resulting model is a very large nonlinear integer programming problem, which is solved by the combination of the following techniques: (i) a disjunctive formulation transforms the non linearities into linear integer constraints; (ii) an heuristic ranking procedure is used to select the most critical combinations of dispatch and contingency scenarios, to be incorporated into the "global" optimization model; afterwards, a "greedy" optimization scheme produces the required reinforcements for the remaining scenarios; (iii) a “horizon year”/backward optimization approach is used to decompose the multi-stage problem into a sequence of one-stage problems. The application of these techniques is illustrated for two "real life" planning studies for the systems of El Salvador and Venezuela.

The optimal hourly scheduling of generation and transmission resources over the next day or week is a key function of both liberalized and centrally planned power sectors. The main difficulty in solving this short-term scheduling (STS) problem lies in the joint modeling of nonlinearities (for example, head variation in hydro plants and quadratic circuit losses); integer decisions/nonconvexities (e.g unit commitment); and time- and space-coupling constraints (such as the water balance in reservoirs and transmission network equations). Although several techniques, in particular Lagrangian Relaxation (LR), have been successfully applied to the solution of STS problems, some limitations appear when a large number of constraints has to be relaxed; also, the LR multiplier updating scheme often has to be “tuned” for each particular power system, thus reducing its flexibility. The approach presented in this paper is based on the transformation of STS nonlinearities and nonconvexities into piecewise mixed linear integer (MILP) constraints. This approach was found to be flexible, allowing the modeling of complex features of both hydrothermal generation and the transmission network. Also, by taking advantage of recent advances in commercial solver capabilities, the MILP scheme was found to be computationally efficient, as illustrated in case studies with seven countries in Latin America and Europe.